# Full text of "Mathematical And Physical Papers - Iii"

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```ON  THE MOTION  OF PENDULUMS.
107
have nothing to do with viscosity*.    Poisson does not give the slightest hint as to the grounds on which he rested his opinion.
69. I come now to the experiments of Dubuat, which are contained in an excellent work of his entitled Principes d'Hydrau-lique, of which the second edition was published in 1786. The first edition does not contain the experiments in question. Dubuat justly remarked that the time of oscillation of a pendulum oscillating in a fluid is greater than it would be in vacuum, not only on account of the buoyancy of the fluid, which diminishes the moving force, but also on account of the mass of fluid which must be regarded as accompanying the pendulum in its motion; and even determined experimentally the mass of fluid which must be regarded as carried by the oscillating body in the case of spheres and of several other solids. Thus Dubuat anticipated by about forty years the discovery of Bessel; but it was not until after the appearance of BesseFs memoir that Dubuat's labours relating to the same subject attracted attention.
Dubuat's method was as follows. Imagine a body suspended by a fine thread or wire and swung in vacuum, and let a be the length of the pendulum, reckoned from the centre of suspension to the centre of oscillation. Now imagine the same body swung in a fluid, in which its apparent weight is p, so that if P denote the weight of fluid displaced, the true weight of the body will be p -h P. Since the moving force is diminished in the ratio of p + P to p, if the inertia of the body were all that had to be overcome, it would be necessary to diminish the length of the pendulum in the same ratio, in order to preserve the same time of oscillation. But since the mass in motion, consists not only of the mass of the body itself, but also of that of the fluid which it carries with it, the pendulum must be shortened still more, in order that the time of oscillation may be unaltered. Let I be the length of the pendulum so shortened, and tt (which for the same reason as before I write instead of Dubuat's n,) a factor greater than unity, such that p + ttP is the weight of the mass in motion ; then
i                         'P   &
whence n--L-
.(167).
* -Journal tie VEcole Poll/technique, Tom. xin. p. 95.```